CVAE is trained to maximize the conditional likelihood \(p(y|x)\) , which involves an intractable marginalization over the latent variable \(z\) . Instead, following [1]} ([1]}), a practical way is to introduce another deep network (parameterized by \(\phi \) ) \(q_{\phi }(z|x,y)\)  to approximate the true posterior distribution \(p(z|x,y)\)  and maximize the evidence lower bound (ELBO) of the log-likelihood function:
\(\vspace*{0.0pt}\begin{split}L^{ELBO}(\theta , \phi )=&\mathbb {E}_{q_{\phi }(z|x, y)}\mathrm {log}(p_{\theta }(y|x, z))-\\&\mathrm {KL}(q_{\phi }(z|x, y)||p_{\theta }(z|x))\\&\le \mathrm {log}p(y|x)\end{split}\) 
